The decibel explained (LONG)
Bill Vermillion
->In <17...@s5.Morgan.COM> se...@fido.morgan.com (Seth Breidbart) writes:
->The decibel (dB) is a unit for expressing a _ratio_ of two signal levels.
->It is not meaningful to say "this signal has a level of 53 dB", because
->that's a bit like saying "this piece of wood is 3.5 long." 3.5 what?
Bad example. Everyone knows that wood is measure on the "log" scale :-)
--
Bill Vermillion - UUCP: uunet!tarpit!bilver!bill
: bi...@bilver.UUCP
Andrew Bettison
>believed the following:
>
> [ misunderstanding about meaning of dB ]
This sort of thing seems to crop up about once every two months or so.
I'll have a go this time...
The decibel (dB) is a unit for expressing a _ratio_ of two signal levels.
It is not meaningful to say "this signal has a level of 53 dB", because
that's a bit like saying "this piece of wood is 3.5 long." 3.5 what?
The decibel is a logarithmic scale, but to explain why we need the dB,
and don't just use straight ratios, a bit of background is needed.
In the natural world, audible phenomena may emit sound pressure waves
with pressure variations ranging from millionths of a Pascals (S.I.
units of air pressure) to millions of Pa or more. This represents
about 12 orders of magnitude. When using electronics to carry audio,
sound pressure is represented by electrical power. The electrical
power present in a conductor may range from microwatts to kilowatts,
again 12 or more orders of magnitude.
It would be a real hassle to express relative signal levels in the form
"ten times power", "one millionth power", etc. And writing it would be
a pain: 2.4e7, 3.2e-8, etc. Worst of all, our perceptual psychology
seems to be arranged non-linearly, so an expression like "ten thousand
times power" doesn't easily translate into anything readily intuitive.
(In fact, it means approximately the difference between talking and
shouting.)
Without going into its origins, the decibel is used for convenience.
Multiplying power by ten means adding 10 dB. Multiply power by one
thousand and add 30 dB. Double power, add 6 dB. One millionth power,
subtract 60 dB. And so on. The mathematical definition of the decibel
difference between two signals with power P1 and P2 is:
10 x log10 ( P1 / P2 )
Now, the decibel cannot be used to quote absolute power levels without
establishing a reference power level with which to compare. There is a
standard measure of power, the dBm, which expresses power as a decibel
ratio against a standard power level of 1 mW. If a signal has a level
of 20 dBm, then it is 100 mW.
The same goes for acoustic volume, also called sound pressure level.
There is a standard unit of SPL, the dBA (A means Acoustic). The sound
pressure is expressed as a decibel value relative to a standard sound
pressure level, which corresponds to an arbitrarily-chosen _very_ quiet
sound (to human ears). A brief table of dBA values and rough
correspondence to familiar environmental sound levels is as follows:
0 dBA generally accepted as the lowest limit of
human audio perception; you're unlikely ever to
hear anything this quiet because of internal
body noises
10 dBA a _very_ quiet room (no wind, cars, etc.) like
a well-insulated recording studio
20 dBA outside in the country side with no wind or
animals
30 dBA inside a library
40 dBA someone talking quietly in the same room
50 dBA someone talking at a distance of a metre or so;
a quiet orchestral passage
60 dBA in a car, with windows down, in the city;
playing a guitar
70 dBA somebody shouting nearby, outside
80 dBA shouting in the same room; a loud orchestral
passage
90 dBA machine-shop floor, drills and machines going;
propeller aircraft engines nearby;
pleasantly loud music
100 dBA someone shouting close to your head; car horns
110 dBA very loud rock concert; commercial jet aircraft
taxiing at 100 m
120 dBA _very_ loud sound: jet engines; jackhammers up close
(threshold of feeling)
130 dBA too loud: explosions; stock car engine up close
(threshold of pain)
140 dBA some sounds have transients up here or beyond;
drums with your ear up close, big explosions
150 dBA can do immediate damage
Three points on this scale need pointing out:
30 dBA is approximately the quietest you'll ever get in the
world without deiberately constructing an insulated room.
Think of it as the environmental noise floor.
120 dBA (or thereabouts) is generally accepted as the
"threshold of feeling". This is the point at which perception
of sound produces a tingly feeling in the ears, or some other
physical sensation in the ear other than just noise. I
sometimes try to describe the sensation as itching in the back
of your cheeks, and wanting to scratch it by curling up your
tongue and rubbing it up against the edges of your palate.
130 dBA or so is the "threshold of pain". This is the point at
which a sound is so loud it is actually painful to the ears.
When talking about the dynamic range of a system, people may say that
it is 65 dB (e.g. normal tape) or 96 dB (e.g. 16-bit digital) without
saying relative to a given signal level. This is okay, because
dynamic range already _is_ a ratio: between the highest and lowest
signal levels representable by the system. The highest signal level is
typically determined by the point at which unacceptable distortion
occurs, and the lowest by the noise floor.
As for why 16 bits per sample yields 96 dB: well, suppose we had, say,
8 bits. If we added another bit to make 9 bits, we'd be able to
represent twice the power as before, without raising the noise floor.
So we'd have an increase of 6 dB in dynamic range. The same would
apply for each extra bit we added. So clearly, each bit of resolution
corresponds to 6 dB of dynamic range. Hence, 16 bits == 96 dB, in a
theoretical sense.
Because there's only about 90 dB between the environmental noise floor
and the threshold of feeling (mentioned above), 16-bit digital, with
its 96 dB of dynamic range is mostly adequate for recording all sorts
of things. There are problems with transients, however, as the attack
transient of a snare drum, for example, may well reach 140 dBA or more,
but so instantaneously that it doesn't cause pain. It will, however,
cause the blink reflex with which you may be familiar: someone hits a
drum close by, and you involuntarily blink. Most recordings fail to
capture the transients necessary to reproduce this effect, but that's
largely academic, because you'd need millions of Watts, and
phenomenally stiff and light speaker transducers to reproduce necessary
the sound pressure level even if it _was_ accurately recorded.
I hope this helps.
--
Andrew Bettison - Softway Pty Ltd
Phone +61-2-698-2322 Internet and...@softway.sw.oz.au
Fax +61-2-699-9174 UUCP uunet!softway.sw.oz.au!andrewb
John Kessenich
>Double power, add 6 dB.
Oops:
10 x log (2) = 3.010...
>As for why 16 bits per sample yields 96 dB: well, suppose we had, say,
>8 bits. If we added another bit to make 9 bits, we'd be able to
>represent twice the power as before, without raising the noise floor.
Should represent twice the amplitude, which corresponds linearly with
voltage. Power varies quadratically with amplitude.
>So we'd have an increase of 6 dB in dynamic range.
Because by adding a bit, you doubled the amplitude, which quadrupled
the power, giving two power doublings, or 2*3.010... dB = 6.020... dB.
>Hence, 16 bits == 96 dB, in a theoretical sense.
Right answer. Incorrect derivation. Bottom line is that
10 x log ((2^16)^2) ~= 96.33
It was still a good posting.
-------------------------
John "not a EE" Kessenich